injective hulls of monars over inverse semigroups

Journal of Algebra 378 (2013) 207–216

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Injective hulls of monars over inverse semigroups

Boris M. Schein

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 January 2012Available online 21 January 2013Communicated by Louis Rowen

To the memory of my mother and teacherSosya Issarov

Keywords:Semigroup actionsInverse semigroupsInjective hulls

A monar is an algebra with unary operations that representsa semigroup acting on a set. Inverse semigroups have “sufficientlymany” injective monars, and inverse semigroup monars haveinjective hulls. These injective hulls are described.

© 2013 Elsevier Inc. All rights reserved.

Foreword

The main theorem of this paper (Theorem 1.2 in Section 1) characterizes injective hulls of monars(equivalently, S-acts, S-systems, S-operands, S-automata, S-semimodules, etc.) over inverse semi-groups. In the extremely specific case when the inverse semigroup S is a linearly ordered semilatticeisomorphic to the chain of rational numbers, our construction becomes the famous Dedekind com-pletion of this chain, that is, the chain of real numbers. Therefore, our construction is a broadgeneralization of the Dedekind one. The proof of the main theorem is given in Section 2. This pa-per is a sequel to the author’s previous work [7], where injective monars over inverse semigroupswere characterized.

A historical remark. The main theorem was announced in the abstracts of my talks at various meet-ings in the then-USSR and USA but the publication of its proof had to be postponed because of“meta-mathematical” circumstances. The main results of this paper were obtained in 1973–1974 andwere a part of my talk at the 13-th All-Union Symposium on General Algebra held in Gomel, formerUSSR, in 1975 (see [5]). At the invitation of Professor L. A. Skornyakov, then the editor in algebra of thejournal Matematicheskiı̆ Sbornik published in Moscow, I began writing a paper after the symposium.

E-mail address: [email protected].

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208 B.M. Schein / Journal of Algebra 378 (2013) 207–216

However, the new Editor-in-Chief appointed by the end of 1975 introduced new policies and replacedmany members of the Editorial Board. It became clear that, regardless of mathematical merits ordemerits of my paper, it would not be published by the journal because of the non-Russian soundof my name. I stopped writing and left the manuscript unfinished. Its two main results appearedwithout proofs in [5] and also, in the form of an abstract, in [6].

In 1976 I was invited to speak at an international meeting on semigroups in Szeged, Hungary.I was not allowed to exit the USSR and so could not participate. Yet Professor G. Pollák, the editor ofits proceedings, asked me to submit a paper based on my undelivered talk. I had a very limited timefor writing my paper and resolving the main difficulty (smuggling the manuscript out of the USSR).This was why I translated from Russian to English only the beginning of my original unfinished paper:the first of the two main results announced in Gomel together with corollaries. It went to Hungaryon January 17, 1979 and appeared a few months later as [7]. In the end of 1979 I was permitted toemigrate from the USSR but not allowed to take manuscripts or any other written materials. I leftsome of them in Moscow at the home of Professor Gregory Freiman.

In the eighties Professors László Márki and Richard Wiegandt visited Professor Freiman, rescuedsome of my materials, and brought them to Budapest. Later Professor Hans Kaiser visited Budapest,took them to Vienna and mailed them to me from there. Thus I recovered my unfinished Russianmanuscript. I had some—rather loose—plans of publishing it and, in 1988, wrote a part of the follow-ing paper in English but was distracted by other projects. So I just gave colloquium talks on theseresults at a few universities in the US and abroad. The last straw that precipitated this paper wasan appeal to publish its result received in September 2005 from Professor Kunitaka Shoji, who wasfamiliar with this result for years.

I am grateful to all the mathematicians mentioned for their help, interest and encouragement.Meanwhile, solutions of similar problems for very considerably narrower classes of inverse semi-

groups were published by other authors.

For completeness’ sake and also because [7] could not be published in a regular mathematicaljournal, I have included a few definitions from [7] in Section 0.

0. Preliminaries

0.0. A unary operation f on a set A is any transformation (that is, a self-map) f : A → A. If a unaryoperation f is applied to an element a ∈ A, the result is written as af . A monar (or a unary alge-bra) on A is a universal algebra of the form A = (A; ( f i)i∈I ), where A is the set of elements of A,I an arbitrary index set, and f i unary operations on A, each of them indexed by an element of I . Thusa monar is a universal algebra that has unary operations only.

Let S be a groupoid (that is, a set with a binary operation written multiplicatively). If s, t ∈ S thenst denotes their product in S .

A monar over S is a monar A = (A; ( f s)s∈S ) with unary operations f s indexed by elements of Ssuch that, for any s, t ∈ S , A satisfies the identity

xfs ft = xfst . (0)

In other words, afs ft = afst holds for all a ∈ A, that is, the composition of the transformations f s

and ft of A is f st .By definition, the class MS of all monars over S is a variety of unary algebras (that is, a class of

all unary algebras that satisfy a certain set of identities).In what follows, we denote the operations f s in the same way as the indexing element s of S , and

so instead of (A; ( f s)s∈S ) we write (A; S) and instead of afs we write as. Thus the identities (0) canbe rewritten in the form

(xs)t = x(st). (1)

B.M. Schein / Journal of Algebra 378 (2013) 207–216 209

0.1. Let (A; S) = (A; ( f s)s∈S ) be a monar over a groupoid S . Clearly, the indexed family ( f s)s∈S isa (homomorphic) representation s �→ f s of S by transformations of the set A. Conversely, supposethat F is such a representation. Writing F in the form of an indexed family ( f s)s∈S with f s = F (s) weobtain a monar (A; ( f s)s∈S ) over S . Therefore the concepts of monars over S and representations of Sby transformations are equivalent.

Every representation F of S is a homomorphism F : S → T (A) of S into the full transformationsemigroup T (A) of all transformations of a suitable set A. It follows that the homomorphic imageF (S) of S is a semigroup.

If E is the least semigroup congruence on the groupoid S then we can replace the monars over Sby those over the quotient semigroup S/E because there is a natural one-to-one correspondencebetween the monars over S and those over S/E .

Thus, without loss of generality, we may (and will) restrict our consideration to monars over semi-groups only.

0.2. Next suppose that A is a right S-act (which is also called an S-operand, S-system, S-polygon,S-automaton, etc.). In other words, we are given a mapping f : A × S → A with ((a, s) f , t) f = (a, st) ffor all a ∈ A and s, t ∈ S . If the second variable s in the mapping (a, s) �→ (a, s) f is fixed, we obtaina mapping afs = (a, s) f of A. Clearly, (A; ( f s)s∈S ) is a monar over S , and conversely, every monarover S corresponds in the above way to a right S-act of S on A.

Analogously defined left S-acts correspond to monars over Sop , the semigroup opposite to S (i.e.s ∗ t = t · s for all s, t ∈ S , where ∗ is the multiplication in Sop and · the multiplication in S). It followsthat the concepts of a monar over S and of an S-act are equivalent.

0.3. The idea of a monar over a semigroup was first introduced in [4]. Monars, S-acts and repre-sentations, while essentially equivalent, look at the same concept from different angles. So why dowe use “monars” rather than the more familiar S-acts or S-sets? The answer is that the class MS ofall monars over a fixed semigroup S is a variety of algebras, and hence all standard definitions andresults of universal algebra and varieties of algebras are immediately applicable. In particular, subal-gebras (submonars in this case) correspond to “subacts”, homomorphisms of monars correspond to“representative homomorphisms” (introduced in [8] in 1956), etc. In particular, there is no need ina special definition of injective S-acts. Instead, we use the standard definition of injective algebras ina variety. Recall that an algebra I from a variety V of algebras is called injective in V if, for everyA,B ∈ V, every homomorphism f : A → I and every embedding (that is, an injective homomor-phism) i :A→ B there exists a homomorphism f ∗ : B → I that makes the diagram

A i

f

B

f ∗

I

commutative.In particular, if C is a class of algebras, then an algebra A ∈ C is called injective in C if it is injective

in the category formed by all algebras from C and all possible homomorphisms between them. Forexample, a monar (A; S) is injective whenever it is injective in the variety MS of all monars over S .

The concept of injectivity was originally introduced for modules over rings in 1940 by ReinholdBaer, and it turned out to be interesting for other algebraic systems. Since the concepts of a moduleover a ring and a monar over a semigroup are quite similar, injective monars form an interestingobject, and they have been considered by various authors.

An injective monar B = (B; S) is called an injective hull (or an injective envelope) of a monar A =(A; S) with respect to an embedding i : A → B if, for every embedding j : A → C of the monar Ainto any injective monar C , there exists an embedding k : B → C of the monars such that j = ik,that is, the diagram


A i

j

B

k

C

is commutative. In other words, the injective hull of A is the “smallest” injective monar that containsan isomorphic copy of A. It follows that if an injective hull of A exists, it is uniquely determined upto an isomorphism over A.

It is known (see [2]) that every monar over every semigroup has an injective hull. In [2] this factwas proved for S-acts, and therefore it required a proof. Since monars form a variety of universalalgebras, this property of monars becomes an obvious corollary to simple observations about varietiesof unary algebras [1].

0.4. If a semigroup S contains an identity 1 and a monar A = (A; S) satisfies the identity x1 = x,then A is called unital. In this case M1

S designates the variety of all unital monars over S .Now suppose that 1 is an element not belonging to S . Let S1 denote the semigroup obtained

from S by adjoining 1 to S as an identity (regardless of the fact whether S has or does not haveits own identity). Notice that we use the designation S1 in a sense slightly different from that usedelsewhere in semigroup literature. Usually, if S already has an identity element, then S1 = S by def-inition. However, in this paper S1 �= S for all S . Even if S has an identity element, we adjoin a newidentity element to it to obtain S1.

If we suppose that a · 1 = a for all a ∈ A, then A1 = (A; S1) is a unital monar over S1.

0.5. Suppose that A = (A; S) is a monar over a semigroup S with identity 1. Then A1 = A · 1 ={a · 1 | a ∈ A} is a submonar of A. Clearly, A1 is unital and the mapping a �→ a · 1 is a homomorphismof A onto A1.

The following result was proved as Theorem 0.7 in [7].

Theorem. A monar A over a semigroup S with identity is injective in the variety MS of all monars over S ifand only if the monar A1 is injective in the variety M1

S of all unital monars over S. In particular, a unital monarover S is injective in MS if and only if it is injective in M1

S .

0.6. Here are some facts about inverse semigroups used later. If S is a semigroup and s, t ∈ S , then tis called an inverse of s whenever sts = s and tst = t . A semigroup S is called inverse if, for every s ∈ S ,there exists a unique inverse of s (it is usually denoted as s−1). Then (ss−1)2 = (ss−1s)s−1 = ss−1

and, analogously, (s−1s)2 = s−1s, that is, ss−1 and s−1s are idempotents for every s in an inversesemigroup S .

Let E be the set of all idempotents of an inverse semigroup S so that E1 is the set of allidempotents of S1. It is known that E is a commutative subsemigroup of S and, for every s ∈ S ,sE = Es = sE1 = E1s.

If we define s � t for s, t ∈ S to mean s ∈ t E , then � is a (partial) order relation on S . It is calledthe natural (or canonical) order.

Inverse semigroups are the most important and most studied class of semigroups beside groups,and investigation of monars over inverse semigroups is both justified and natural.

In the sequel all semigroups are supposed to be inverse!

0.7. The following facts were established and definitions introduced in [7]:

Lemma. Define a � b if a ∈ bE1 for a,b ∈ A in a monar A. Then � is a (partial) order relation stable under alloperations of A (that is, for all a,b ∈ A and s ∈ S, a � b ⇒ as � bs). The order relation � is called natural.


0.8. A subset B of A is called a tail (or minorantly closed) if a ∈ A and b � a imply b ∈ A for alla,b ∈ A. In other words, B is a tail exactly when B E ⊆ B (or, equivalently, B E1 = B).

A subset B of A is called compatible if there exists a mapping B → E1 (b �→ eb for every b ∈ B)such that beb = b and b1eb2 = b2eb1 for all b,b1,b2 ∈ B . We call it a compatibility mapping. For exam-ple, the empty subset ∅ and all one-element subsets {a} of A are compatible.

A compatible tail is a compatible subset that is a tail.The following lemma was proved in [7] as Lemma 1.10.

Lemma. If B is a compatible tail of A with a compatibility mapping b �→ eb, then Beb = bE1 .

0.9. If B ⊆ A,a ∈ A and B ⊆ aE1 (that is, b � a for all b ∈ B), then a is called an upper bound of B .A minimal upper bound of B is any minimal (with respect to �) element of the set of all upper boundsof B .

An element a is called a face of B , if

(i) a is an upper bound of B and(ii) for all s, t ∈ S1, Bs = Bt implies as = at .

A historical remark. The term “face” was originally coined in [7]. In the 1973–1974 Russianmanuscript I used the word “gran�.” In Russian it means both “bound” (as in “upper bound”) and“face” (of a polyhedron or a gem) and so “the least upper bound” sounds in Russian as “the leastupper face”. I adopted “gran�” as used in ordered set contexts. Rewriting a part of my manuscript inEnglish I replaced “gran�” by “face” because my “face” is a minimal upper bound, and the word hasno order-theoretical connotations, had not been used in ordered set context, and is unambiguous, allthese virtues not shared by “bound”.

In 1991 two foreign authors whose native language is neither English nor Russian published in theUS a joint paper devoted to a simple corollary of my second main result. They followed [7] almostverbatim and also used my term “face” and the diagrams and examples from my paper. So inspiredthey were that they had never mentioned [7] nor its author, nor made any indications that theseresults might not have been their own. This is the sincerest form of flattery.

The following lemma was proved in [7] as Lemma 1.7 and Example 1.8.

Lemma.

(i) If a subset B has an upper bound, then B is compatible.(ii) If a is a face of B then as is a face of Bs for all a ∈ A and s ∈ S1 .

(iii) Each face of B is a minimal upper bound of B.(iv) Distinct faces of B are never compatible one with the other.(v) If B has the greatest element b, then b is the only face of B. Such a face is called trivial.

0.10. Now we clarify the connection between the least upper bounds and faces in a monar.It is not difficult to see that even if a subset B of a monar A has the least upper bound

∨B in A,

yet B may possess no face.We say that a is the distributive least upper bound (d.l.u.b.) of B if as is the least upper bound of Bs

for every s ∈ S1. In other words, (i) the least upper bound∨

(Bs) of Bs exists for every s ∈ S; (ii) a isthe least upper bound

∨B of B , and (iii) (

∨B)s = ∨

(Bs).Observe that if a is the d.l.u.b. of B and Bs = Bt for some s, t ∈ S1, then

as =(∨

B)

s =∨

(Bs) =∨

(Bt) =(∨

B)

t = at.

Therefore, a is a face of B .


Analogously, we say that a is a distributive minimal upper bound (d.m.u.b.) of B if as is a minimalupper bound of Bs for every s ∈ S1 and if Bs = Bt then as = at for all s, t ∈ S1.

Lemma.

(i) If B possesses both the least upper bound and a face, then the least upper bound of B is its only face.(ii) An element a ∈ A is a face of a subset B ⊆ A if and only if a is a distributive minimal upper bound of B.

Proof. (i) If a is the l.u.b. and b a face of B then a � b because b is an upper bound and a the leastupper bound of B . By Lemma 0.9(iii), b is a minimal upper bound of B and a � b implies a = b.It follows that a is the only face of B .

(ii) We have already seen that a d.m.u.b. is a face.Conversely, if a is a face of B then, by Lemma 0.9(ii), as is a face of Bs and, by part (iii) of the

same lemma, as is a minimal upper bound of Bs. If Bs = Bt for some s, t ∈ S1, then as = at by thedefinition of face. So a is a d.m.u.b. of B . �0.11. The following theorem is the main result of [7].

Theorem. A monar A over an inverse semigroup S is injective if and only if every compatible subset of A hasa face.

1. Main theorem

1.0. A subset B ⊆ A is called a filter if B is a compatible tail and, for every e ∈ E1 for which Be hasa face in A, one of the faces of Be belongs to B .

As part (ii) of Lemma 1.1 will show, if B is a filter and Be has a face, then Be contains the largestelement that is the only (trivial) face of Be.

Lemma. If B is a filter of A, then Bs is a filter for every s ∈ S.

Proof. Suppose that B is filter. Since B is compatible, Bs is compatible by Lemma 1.6 of [7].Also, BsE1 = B E1s = Bs because sE1 = E1s in every inverse semigroup and B E1 = B . Thus Bs is a

tail.Let Bse have a face a for some e ∈ E1. By Lemma 1.7 of [7], as−1 is a face of Bses−1. However,

ses−1 ∈ E , hence Bses−1 has a face, say b, that belongs to B and so bs ∈ Bs. By Lemma 1.7 of [7], as−1

is a face of Bses−1s = Bse (because es−1s = s−1se and ss−1s = s so that ses−1s = se). Thus, for everye ∈ E1, if Bse has a face, then one of its faces belongs to B . It follows that Bs is filter. �1.1. Let F(A) denote the set of all filters of A. By Lemma 1.0, F(A) is a monar over S , in whichf s(B) = Bs for every s ∈ S and B ∈F(A).

Lemma.

(i) If a ∈ A then aτ = aE1 = {b ∈ A | b � a} is a filter in A. We call it the principal filter generated by a.(ii) If B is a filter in A, e ∈ E1 and Be has a face, then this face is trivial. That is, Be = aτ for some a ∈ B.

Proof. (i) By our definition of τ , aτ is a tail because (aτ )E1 = aE1 E1 = aE1 = aτ . Also, for everyb ∈ aτ , there exists eb ∈ E1 such that b = aeb . Thus beb = aebeb = aeb = b and, for every b1,b2 ∈ aτ ,we have b1eb2 = aeb1 eb2 = aeb2 eb1 = b2eb1 . So aτ is compatible.

Also, (ae)τ = (ae)E1 = a(eE1) = aE1e = (aτ )e so that (aτ )e = (ae)τ for each e ∈ E1. Thus ae is thegreatest element of (aτ )e. By Lemma 0.9(v), ae is the trivial face of (aτ )e. Since ae ∈ (aτ )e, we seethat aτ is a filter.


(ii) Suppose that Be has a face. Since B is a filter, then Be has a face a belonging to B . It followsfrom Be = Bee that a = ae ∈ Be, that is, a is the greatest element of Be. Obviously, Be is a tail, andhence Be = aτ and, by Lemma 0.9(v), a is the trivial face of Be. �

Therefore, τ is a mapping of A into F(A). As proved in Lemma 1.9 in [7], τ is an isomorphicembedding of A into F(A).

1.2. Now we can state the main result of this paper.

Main Theorem. If A is a monar over an inverse semigroup S, then F(A) is the injective hull of A with respectto the embedding τ :A→F(A).

Remark. Every monar A = (A; S) is an ordered set. Observe that its order relation � was definedusing the reduct (A; E S ) of A.

The semilattice E S defines an order on A such that each subset aτ is a semilattice isomorphicto the principal ideal E S (aa−1) of E S . Indeed, if a1,a2 ∈ aτ then a1 = ae1 and a2 = ae2 for certaine1, e2 ∈ E S and the meet a1 ∧a2 = ae1e2 = a[(aa−1e1)∧(aa−1e2)]. Thus the monar A considered as theordered set (A;�) is a union of semilattices isomorphic to all principal ideals of the semilattice E S .

Section 1.3 shows that our construction of F(A) is a broad generalization of the Dedekind–MacNeille completion.

1.3. Dedekind completion. If A = (A;�) is a chain (that is, a set A linearly ordered by �) we canconsider it as a (lower) semilattice with the operation a1a2 = min{a1,a2}. Thus A becomes an in-verse semigroup. Turn A into a monar over the inverse semigroup A, where A “acts” on A by righttranslations: if a ∈ A then xfa = min{x,a} for every x ∈ A. What are the filters of the monar A?

Every subset B of A is compatible. Indeed, for every b ∈ B define eb = b (the compatibility map-ping is just the identity mapping). Then bec = min{b, c} = ceb and beb = b. Therefore, the compatibletails are just tails. Let B be a filter different from ∅ and A. If a is a face of B then a is the leastupper bound of B , and hence it is the only face of B . Also, a ∈ B because B is a filter. Thus B = aτand B is a principal filter. Otherwise, B has no faces, and hence B has no l.u.b. Observe that ∅ isa filter precisely when the chain A has no least element. Indeed, ∅ is a compatible tail and the leastelement e of A (if it exists) would be its l.u.b., and hence ∅ is a filter precisely when e ∈∅, which isa contradiction. Analogously, A is a filter precisely when of A has the greatest element.

Let A be the chain (Q;�) of rational numbers ordered in the normal way. For each filter B of Qlet Bc be the complement of B . Then a ∈ Bc precisely when b � a for all b ∈ B . This decomposition ofQ into (B, Bc) is exactly the “Dedekind cut” on Q. The set of all Dedekind cuts is identified with theset R of all real numbers to which two more cuts −∞ = (∅,Q) and ∞ = (Q,∅) are adjoined as the“endpoints”. Thus the chain R of real numbers supplied with −∞ and ∞ is the injective hull of Qwith respect to the embedding τ : Q → R.

Instead of Q we could have taken any other chain considered as a semilattice and prove that itsinjective hull is its Dedekind completion.

Another generalization of the Dedekind construction (that is called the Dedekind–MacNeille com-pletion in [3]) is likewise a very special case of our construction. We do not consider it here becauseit is a completion of an arbitrary ordered set that does not have to be a semilattice and thus aninverse semigroup. However, we can easily embed an ordered set into a semilattice in a “canonicalway” and then consider the injective hull of this semilattice thus obtaining the MacNeille completion.

2. Proof of the Main Theorem

2.0. Let C(A) be the set of all compatible tails of A. By Lemma 1.6 of [7], C(A) is a monar in whichB fs = Bs for every B ∈ C(A) and every s ∈ S . Clearly, F(A) is a submonar of C(A), and hence τ isan embedding of A in C(A).


Lemma. C(A) is an injective monar.

Proof. We prove that every compatible subset of C(A) has a face and apply Theorem 0.11 to concludethat C(A) is an injective monar.

Suppose that {Bi}i∈I is a compatible subset of C(A) and B = ⋃{Bi}i∈I . We claim that B ∈ C(A)

and B is a face of {Bi}i∈I in C(A).Since {Bi}i∈I is compatible, then for every i ∈ I there exists ei ∈ E1 such that Biei = Bi and Bie j =

B jei for all i, j ∈ I . Also Bi E1 = Bi for all i ∈ I . Therefore, B E1 = ⋃Bi E1 = ⋃

Bi = B and B is a tail.For every i ∈ I , Bi is compatible in A. So there exists a family {eib}b∈Bi of elements of E1 such that

beib = b and beic = ceib for all b, c ∈ Bi .For each b ∈ B choose i ∈ I such that b ∈ Bi and define fb = eibei . It follows from Bi = Biei that,

for every b ∈ Bi , there exists c ∈ Bi such that b = cei . Therefore, bei = ceiei = ciei = b, and hencebfb = bebiei = bei = b.

Suppose that b1,b2 ∈ B, fb1 = eib1 ei and fb2 = e jb2 e j , where b1 ∈ Bi and b2 ∈ B j . Then b1e j ∈Bie j = B jei , that is, b1e j = cei for some c ∈ B j . It follows that b1 fb2 = b1e jb2 e j = b1e je jb2 = ceie jb2 =ce jb2 ei = b2e jcei � b2. Therefore, b1 fb2 = b1 fb1 fb2 = b1 fb2 fb1 � b2 fb1 and, analogously, b2 fb1 � b1 fb2 ,that is, b1 fb2 = b2 fb1 . Thus B is compatible in A.

Also, B jei ⊆ B j E1 = B j and B je j = B j for all j ∈ I . Thus Be j = ⋃i∈I Bie j = ⋃

i∈I B jei = B j andB j � B in C(A). It follows that B is an upper bound of the subset {Bi}i∈I ⊆ C(A).

Suppose that {Bi}i∈I s = {Bi}i∈I t for some s, t ∈ S1, that is, for every i ∈ I , there exist j,k ∈ I suchthat Bi s = B jt and Bit = Bks. Then Bs = ⋃

i∈I Bi s = ⋃j∈I B jt = Bt . Therefore, B is a face of {Bi}i∈I .

It follows that every compatible subset of C(A) has a face (and even the greatest lower bound!).By Theorem 0.10, C(A) is an injective monar. �2.1. Lemma. The embedding τ : A → F(A) is essential. In other words, every congruence on F(A) thatinduces the identity congruence on A is the identity congruence on F(A).

Proof. Recall that a congruence on a universal algebra A is an equivalence relation ε that is a sub-algebra of the algebra A × A. Thus a congruence on a monar A is any equivalence relation ε on Asuch that, for all a1,a2 ∈ A and all s ∈ S , a1 ≡ a2 implies a1s ≡ a2s.

Let ≡ be a congruence on F(A). It induces the congruence ≡ on A such that a1 ≡ a2 ⇔ a1τ ≡ a2τfor all a1,a2 ∈ A. Here we denote the congruences on F(A) and on A by the same symbol ≡.

Suppose that the congruence ≡ on F(A) induces the equality relation on A, that is, (∀a1,a2 ∈ A)

[a1τ ≡ a2τ ⇒ a1 = a2]. Our goal is to prove that ≡ is the equality relation on F(A).Let B ≡ C for some B, C ∈ F(A). Here B is compatible, and hence, for each b ∈ B , we can choose

eb ∈ E1 such that conditions 0.8 hold. Analogously, for each c ∈ C , choose fc ∈ E1 so that the obviousanalogs of conditions 0.8 (when e is replaced by f ) hold. By Lemma 1.10 of [7], Beb = bτ . It followsthat bτ = Beb ≡ Ceb . It is also clear that

(bfc)τ = (bτ ) fc = Beb fc ≡ Ceb fc = C fceb = (cτ )eb = (ceb)τ ,

and hence (bfc)τ = (ceb)τ and ceb = bfc � b. Thus b is an upper bound of Ceb .Now let Cebs = Cebt for some s, t ∈ S1. Then (bs)τ = (bτ )s ≡ Cebs = Cebt ≡ (bτ )t = (bt)τ , and

hence (bs)τ = (bt)τ and so bs = bt . It follows that b is a face of Ceb . Also, C ∈ C(A) and so, byLemma 0.8 Ceb = bτ . Thus b ∈ Ceb ⊆ C . Therefore, B ⊆ C . Analogously, C ⊆ B , and hence B = C .It follows that ≡ is the identity relation on F(A) and τ is an essential embedding. �2.2. As observed in 0.3, every monar A has an injective hull I(A) with respect to an embeddingα : A → I(A). An injective hull of A is (up to isomorphism over A) both the greatest essential andthe least injective extension of A.


Thus there exist embeddings β :F(A) → I(A) and γ : I(A) → C(A) such that the diagram

A τ

α

F(A)ι

β

C(A)

I(A)

γ

is commutative. Here ι is the identity embedding of F(A) in C(A). Therefore, (I(A))γ is the injectivehull of A with respect to the embedding αγ = τβγ = τ ι. Without loss of generality, we can identifyI(A) and (I(A))γ , that is, we can assume that A τ→F(A) ⊆ I(A) ⊆ C(A).

To complete the proof we need to verify the equality F(A) = I(A). Indeed, suppose that D ∈I(A) and d is a face of D in A. Consider a binary relation ρ on I(A) defined as follows. For anyB, C ∈ I(A), BρC if and only if one of the four possibilities holds:

(1) B = C ;(2) B = Ds and C = (ds)τ for some s ∈ S1;(3) B = (ds)τ and C = Ds for some s ∈ S1;(4) B = Ds and C = Dt for some s, t ∈ S1 for which ds = dt .

We show that ρ is a congruence on I(A) (it is the congruence generated by the relation D ≡ dτ ).The reflexivity and symmetry of ρ follow from its definition. To see that ρ is transitive suppose

that BρC and Cρ F for some B, C, F ∈ I(A).

Case 1. If B = C or C = F then Bρ F . Therefore, we assume that B �= C and C �= F .

Case 2. Let B = Ds and C = (ds)τ for some s ∈ S1. If C = Dt and F = (dt)τ , then (ds)τ = C = Dt .By Lemma 0.9(ii), dt is a face of Dt = (ds)τ . By Lemma 0.9(v), ds = dt , and hence C = F .

If C = (dt)τ and F = Dt , then (ds)τ = C = (dt)τ , and hence Bρ F by condition (4). If C = Dt andF = Du, where dt = du, then it follows from (ds)τ = C = Dt that, as we have just seen, ds = dt , andhence ds = du and Bρ F .

Case 3. Suppose that B = (ds)τ and C = Ds. If C = Dt and F = (dt)τ , then Ds = Dt implies ds = dtbecause d is a face of D . Therefore, B = F and so Bρ F . If C = (dt)τ and F = Dt then Ds = C = (dt)τimply ds = dt and B = C . If C = Dt and F = Du with dt = du, then Ds = C = Dt imply ds = dt , andhence ds = du and Bρ F .

Case 4. Let B = Ds, C = Dt and ds = dt . If C = Du and F = (du)τ , then Dt = C = Du, dt = du, ds = du,and hence Bρ F . If C = (du)τ and F = Du, then Dt = C = (duτ ) and so dt = du, and hence ds = duand Bρ F . If C = Du, F = D v and du = dv , then Dt = C = Du imply dt = du, and hence ds = dv andBρ F .

Thus ρ is transitive, and hence it is an equivalence relation. It is easily seen from the definitionof ρ that BρC imply BsρC s for all B, C ∈ I(A) and s ∈ S . Therefore, ρ is a congruence on I(A).

Now suppose that b, c ∈ A and (bτ )ρ(cτ ). If bτ = Ds and cτ = (ds)τ , then c = ds and ds is a faceof the subset Ds = bτ . Therefore, c = b. Analogously, we obtain c = b in the case when bτ = (ds)τand cτ = Ds. If bτ = Ds and cτ = Dt with ds = dt , then b = ds and c = dt , whence b = c. Therefore,(bτ )ρ(cτ ) implies bτ = cτ , that is, ρ induces the identity congruence on the image Aτ of the monarA in I(A). Since the injective hull of a monar is its essential extension, ρ is the identity congruence.Thus Dρ(dτ ) implies D = dτ . We have proved that if D ∈ I(A) and D has a face d, then D = dτ .

Now suppose that B is an arbitrary element of I(A). It follows from I(A) ⊆ C(A) that B isa compatible tail in A. Suppose that Be has a face in A for some e ∈ E . As we have just proved,


Be ∈ I(A) implies Be = bτ for some b. Then b is a face of Be and b ∈ Be ⊆ B . Thus B is a filter,that is, B ∈F(A). Therefore, I(A) ⊆F(A). As we have seen earlier, F(A) ⊆ I(A). So I(A) =F(A),which completes the proof of the Main Theorem. �References

[1] B. Banaschewski, Injectivity and essential extensions in equational classes of algebras, in: Proc. Conf. on Universal Algebra,Queen’s Univ., Kingston, Ont., 1969, pp. 131–147.

[2] P. Berthiaume, The injective envelope of S-sets, Canad. Math. Bull. 10 (1967) 261–273.[3] Garrett Birkhoff, Lattice Theory, 3rd edition, Amer. Math. Soc., Providence, RI, 1967.[4] B.M. Xa�n, Predstavlenie polugrupp. Tezisy dokladov 1-go Vseso�znogo Simpoziuma po Teorii Polu-

grupp, Sverdlovsk, 1969, pp. 79–89 [Boris M. Schein, Representation of semigroups, in: Summaries of Talks at the 1stAll-Union Symposium on Semigroup Theory, Sverdlovsk, 1969, pp. 79–89 (in Russian)].

[5] B.M. Xa�n, In�ektivnye monary nad inversnymi polugruppami. Tezisy dokladov Vseso�znogo Al-gebraiqeskogo Simpoziuma, Gomel�, 1975, pp. 241–242 [Boris M. Schein, Injective monars over inverse semigroups,in: Summaries of Talks at the All-Union Algebra Symposium, Gomel, 1975, pp. 241–242 (in Russian)].

[6] Boris M. Schein, Injective S-acts over inverse semigroups, Notices Amer. Math. Soc. 23 (1976) A-640–A-641 (Abstract 76T-A254).

[7] Boris M. Schein, Injective monars over inverse semigroups, in: G. Pollák (Ed.), Algebraic Theory of Semigroups, Szeged (Hun-gary), 1976, in: Colloq. Math. Soc. J. Bolyai, vol. 20, North-Holland Publ. Co., Amsterdam, 1979, pp. 519–544.

[8] V.V. Wagner (V.V. Vagner), Predstavlenie upor�doqennyh polugrupp, Mat. Sb. (N.S.) 38(80) (2) (1956) 203–240 [Representation of ordered semigroups, Mat. Sb. (N.S.) 38(80) (2) (1956) 203–240 (in Russian)]; English translation in:Amer. Math. Soc. Transl. Ser. 2 36 (1964) 295–336.

injective hulls of monars over inverse semigroups

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